Let $F : \mathcal{C} \to \mathcal{D}$ be an equivalence of categories and $(\mathcal{D},\otimes,1)$ be a monoidal structure on $\mathcal{D}$. Then we have an induced monoidal structure on $\mathcal{C}\,$: Choose a quasi-inverse $F^{-1} : \mathcal{D} \to \mathcal{C}$ and isomorphisms $\mathrm{id}_{\mathcal{C}} \cong F^{-1} \circ F$, $\mathrm{id}_{\mathcal{D}} \cong F \circ F^{-1}$ satisfying the triangle identities. For $x,y \in \mathcal{C}$ we define $x \otimes y := F^{-1}(F(x) \otimes F(y))$. Hence, there is an isomorphism $F(x \otimes y) \cong F(x) \otimes F(y)$. We also let $1 := F^{-1}(1)$. The associator is defined by $$x \otimes (y \otimes z) = F^{-1}(F(x) \otimes F(y \otimes z)) \cong F^{-1}(F(x) \otimes (F(x) \otimes F(y)))$$ $$ \cong F^{-1}((F(x) \otimes F(y)) \otimes F(z)) \cong F^{-1}(F(x \otimes y) \otimes F(z)) = (x \otimes y) \otimes z.$$

Let $F : \mathcal{C} \to \mathcal{D}$ be an equivalence of categories and $(\mathcal{D},\otimes,1)$ be a monoidal structure on $\mathcal{D}$. Then we have an induced monoidal structure on $\mathcal{C}\,$: Choose a quasi-inverse $F^{-1} : \mathcal{D} \to \mathcal{C}$ and isomorphisms $\mathrm{id}_{\mathcal{C}} \cong F^{-1} \circ F$, $\mathrm{id}_{\mathcal{D}} \cong F \circ F^{-1}$ satisfying the triangle identities. For $x,y \in \mathcal{C}$ we define $x \otimes y := F^{-1}(F(x) \otimes F(y))$. Hence, there is an isomorphism $F(x \otimes y) \cong F(x) \otimes F(y)$. We also let $1 := F^{-1}(1)$. The associator is defined by $$x \otimes (y \otimes z) = F^{-1}(F(x) \otimes F(y \otimes z)) \cong F^{-1}(F(x) \otimes (F(y) \otimes F(z)))$$ $$ \cong F^{-1}((F(x) \otimes F(y)) \otimes F(z)) \cong F^{-1}(F(x \otimes y) \otimes F(z)) = (x \otimes y) \otimes z.$$